Finite volume schemes for multi-dimensional hyperbolic systems based on the use of bicharacteristics

Mária Lukáčová-Medviďová; Jitka Saibertová

Applications of Mathematics (2006)

  • Volume: 51, Issue: 3, page 205-228
  • ISSN: 0862-7940

Abstract

top
In this paper we present recent results for the bicharacteristic based finite volume schemes, the so-called finite volume evolution Galerkin (FVEG) schemes. These methods were proposed to solve multi-dimensional hyperbolic conservation laws. They combine the usually conflicting design objectives of using the conservation form and following the characteristics, or bicharacteristics. This is realized by combining the finite volume formulation with approximate evolution operators, which use bicharacteristics of the multi-dimensional hyperbolic system. In this way all of the infinitely many directions of wave propagation are taken into account. The main goal of this paper is to present a self-contained overview on the recent results. We study the L 1 -stability of the finite volume schemes obtained by various approximations of the flux integrals. Several numerical experiments presented in the last section confirm robustness and correct multi-dimensional behaviour of the FVEG methods.

How to cite

top

Lukáčová-Medviďová, Mária, and Saibertová, Jitka. "Finite volume schemes for multi-dimensional hyperbolic systems based on the use of bicharacteristics." Applications of Mathematics 51.3 (2006): 205-228. <http://eudml.org/doc/33251>.

@article{Lukáčová2006,
abstract = {In this paper we present recent results for the bicharacteristic based finite volume schemes, the so-called finite volume evolution Galerkin (FVEG) schemes. These methods were proposed to solve multi-dimensional hyperbolic conservation laws. They combine the usually conflicting design objectives of using the conservation form and following the characteristics, or bicharacteristics. This is realized by combining the finite volume formulation with approximate evolution operators, which use bicharacteristics of the multi-dimensional hyperbolic system. In this way all of the infinitely many directions of wave propagation are taken into account. The main goal of this paper is to present a self-contained overview on the recent results. We study the $L^1$-stability of the finite volume schemes obtained by various approximations of the flux integrals. Several numerical experiments presented in the last section confirm robustness and correct multi-dimensional behaviour of the FVEG methods.},
author = {Lukáčová-Medviďová, Mária, Saibertová, Jitka},
journal = {Applications of Mathematics},
keywords = {multidimensional finite volume methods; bicharacteristics; hyperbolic systems; wave equation; Euler equations; multidimensional finite volume methods; bicharacteristics; hyperbolic systems; wave equation; Euler equations},
language = {eng},
number = {3},
pages = {205-228},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Finite volume schemes for multi-dimensional hyperbolic systems based on the use of bicharacteristics},
url = {http://eudml.org/doc/33251},
volume = {51},
year = {2006},
}

TY - JOUR
AU - Lukáčová-Medviďová, Mária
AU - Saibertová, Jitka
TI - Finite volume schemes for multi-dimensional hyperbolic systems based on the use of bicharacteristics
JO - Applications of Mathematics
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 51
IS - 3
SP - 205
EP - 228
AB - In this paper we present recent results for the bicharacteristic based finite volume schemes, the so-called finite volume evolution Galerkin (FVEG) schemes. These methods were proposed to solve multi-dimensional hyperbolic conservation laws. They combine the usually conflicting design objectives of using the conservation form and following the characteristics, or bicharacteristics. This is realized by combining the finite volume formulation with approximate evolution operators, which use bicharacteristics of the multi-dimensional hyperbolic system. In this way all of the infinitely many directions of wave propagation are taken into account. The main goal of this paper is to present a self-contained overview on the recent results. We study the $L^1$-stability of the finite volume schemes obtained by various approximations of the flux integrals. Several numerical experiments presented in the last section confirm robustness and correct multi-dimensional behaviour of the FVEG methods.
LA - eng
KW - multidimensional finite volume methods; bicharacteristics; hyperbolic systems; wave equation; Euler equations; multidimensional finite volume methods; bicharacteristics; hyperbolic systems; wave equation; Euler equations
UR - http://eudml.org/doc/33251
ER -

References

top
  1. 10.1006/jcph.2000.6666, J.  Comput. Phys. 167 (2001), 177–195. (2001) DOI10.1006/jcph.2000.6666
  2. 10.1006/jcph.1998.5959, J.  Comput. Phys. 143 (1998), 181–199. (1998) MR1624688DOI10.1006/jcph.1998.5959
  3. 10.1016/j.compfluid.2003.12.002, Comput. Fluids 34 (2003), 541–560. (2003) DOI10.1016/j.compfluid.2003.12.002
  4. 10.1137/S1064827500373413, SIAM J. Sci. Comput. 23 (2001), 707–740. (2001) MR1860961DOI10.1137/S1064827500373413
  5. Wave propagation algorithms for multidimensional hyperbolic systems, J. Comput. Phys. 131 (1997), 327–353. (1997) Zbl0872.76075
  6. 10.3934/dcds.2003.9.559, Discrete Contin. Dyn. Syst.  (A) 9 (2003), 559–576. (2003) MR1974525DOI10.3934/dcds.2003.9.559
  7. 10.1090/S0025-5718-00-01228-X, Math. Comput. 69 (2000), 1355–1384. (2000) MR1709154DOI10.1090/S0025-5718-00-01228-X
  8. 10.1002/fld.297, Internat. J.  Numer. Methods Fluids 40 (2002), 425–434. (2002) MR1932992DOI10.1002/fld.297
  9. 10.1137/S1064827502419439, SIAM J. Sci. Comput. 26 (2004), 1–30. (2004) MR2114332DOI10.1137/S1064827502419439
  10. Finite volume evolution Galerkin methods for nonlinear hyperbolic systems, J. Comput. Phys. 183 (2002), 533–562. (2002) MR1947781
  11. Lax-Wendroff type second order evolution Galerkin methods for multidimensional hyperbolic systems, East-West  Numer. Math. 8 (2000), 127–152. (2000) MR1773188
  12. On the stability of the evolution Galerkin schemes applied to a two-dimensional wave equation system, SIAM J.  Numer. Anal. (2006), In print. (2006) MR2257117
  13. 10.1137/S0036142997316967, SIAM J.  Numer. Anal. 35 (1998), 2195–2222. (1998) Zbl0927.65119MR1655843DOI10.1137/S0036142997316967
  14. 10.1006/jcph.2000.6598, J.  Comput. Phys. 164 (2000), 283–334. (2000) MR1792514DOI10.1006/jcph.2000.6598
  15. 10.1002/(SICI)1099-1476(19970910)20:13<1111::AID-MMA903>3.0.CO;2-1, Math. Methods Appl. Sci. 20 (1997), 1111–1125. (1997) MR1465396DOI10.1002/(SICI)1099-1476(19970910)20:13<1111::AID-MMA903>3.0.CO;2-1
  16. Multidimensional characteristic Galerkin schemes and evolution operators for hyperbolic systems, PhD. thesis, University Hannover, 1995. (1995) Zbl0831.76067MR1361170
  17. Genuinely multidimensional finite volume schemes for systems of conservation laws, PhD. thesis, Technical University Brno, 2003. (2003) 
  18. Evolution Galerkin schemes and discrete boundary conditions for multidimensional first order systems, PhD. thesis, University of Magdeburg, 2002. (2002) 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.